Problem: Simplify and expand the following expression: $ \dfrac{3x + 8}{x + 5}+\dfrac{5x - 4}{x + 3} $
Answer: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(x + 5)(x + 3)$ Multiply the first term by $\dfrac{x + 3}{x + 3}$ $ \begin{align*} \dfrac{3x + 8}{x + 5} \times \dfrac{x + 3}{x + 3} & = \dfrac{(3x + 8)(x + 3)}{(x + 5)(x + 3)} \\ & = \dfrac{3x^2 + 17x + 24}{(x + 5)(x + 3)}\end{align*} $ Multiply the second term by $\dfrac{x + 5}{x + 5}$ $ \begin{align*} \dfrac{5x - 4}{x + 3} \times \dfrac{x + 5}{x + 5} & = \dfrac{(5x - 4)(x + 5)}{(x + 3)(x + 5)} \\ & = \dfrac{5x^2 + 21x - 20}{(x + 3)(x + 5)}\end{align*} $ Now we have: $ = \dfrac{3x^2 + 17x + 24}{(x + 5)(x + 3)} + \dfrac{5x^2 + 21x - 20}{(x + 3)(x + 5)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{3x^2 + 17x + 24 + 5x^2 + 21x - 20}{(x + 5)(x + 3)} $ $ = \dfrac{8x^2 + 38x + 4}{(x + 5)(x + 3)}$ Expand the denominator: $ = \dfrac{8x^2 + 38x + 4}{x^2 + 8x + 15}$